| Step | Hyp | Ref
| Expression |
| 1 | | minmar1fval.q |
. 2
⊢ 𝑄 = (𝑁 minMatR1 𝑅) |
| 2 | | oveq12 7440 |
. . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅)) |
| 3 | | minmar1fval.a |
. . . . . . . 8
⊢ 𝐴 = (𝑁 Mat 𝑅) |
| 4 | 2, 3 | eqtr4di 2795 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴) |
| 5 | 4 | fveq2d 6910 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴)) |
| 6 | | minmar1fval.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
| 7 | 5, 6 | eqtr4di 2795 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵) |
| 8 | | simpl 482 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁) |
| 9 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) |
| 10 | | minmar1fval.o |
. . . . . . . . . . 11
⊢ 1 =
(1r‘𝑅) |
| 11 | 9, 10 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
| 12 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) |
| 13 | | minmar1fval.z |
. . . . . . . . . . 11
⊢ 0 =
(0g‘𝑅) |
| 14 | 12, 13 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
| 15 | 11, 14 | ifeq12d 4547 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)) = if(𝑗 = 𝑙, 1 , 0 )) |
| 16 | 15 | ifeq1d 4545 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))) |
| 17 | 16 | adantl 481 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))) |
| 18 | 8, 8, 17 | mpoeq123dv 7508 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))) |
| 19 | 8, 8, 18 | mpoeq123dv 7508 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) |
| 20 | 7, 19 | mpteq12dv 5233 |
. . . 4
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗))))) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))) |
| 21 | | df-minmar1 22641 |
. . . 4
⊢ minMatR1
= (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗)))))) |
| 22 | 6 | fvexi 6920 |
. . . . 5
⊢ 𝐵 ∈ V |
| 23 | 22 | mptex 7243 |
. . . 4
⊢ (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) ∈ V |
| 24 | 20, 21, 23 | ovmpoa 7588 |
. . 3
⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))) |
| 25 | 21 | mpondm0 7673 |
. . . . 5
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = ∅) |
| 26 | | mpt0 6710 |
. . . . 5
⊢ (𝑚 ∈ ∅ ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = ∅ |
| 27 | 25, 26 | eqtr4di 2795 |
. . . 4
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚 ∈ ∅ ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))) |
| 28 | 3 | fveq2i 6909 |
. . . . . . 7
⊢
(Base‘𝐴) =
(Base‘(𝑁 Mat 𝑅)) |
| 29 | 6, 28 | eqtri 2765 |
. . . . . 6
⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅)) |
| 30 | | matbas0pc 22413 |
. . . . . 6
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) →
(Base‘(𝑁 Mat 𝑅)) = ∅) |
| 31 | 29, 30 | eqtrid 2789 |
. . . . 5
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅) |
| 32 | 31 | mpteq1d 5237 |
. . . 4
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) = (𝑚 ∈ ∅ ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))) |
| 33 | 27, 32 | eqtr4d 2780 |
. . 3
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 minMatR1 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))) |
| 34 | 24, 33 | pm2.61i 182 |
. 2
⊢ (𝑁 minMatR1 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) |
| 35 | 1, 34 | eqtri 2765 |
1
⊢ 𝑄 = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗))))) |